Finite-Element Modeling Step By Step

Nov. 15, 2002
Software for performing meaningful analysis increasingly shows up on the desktops of design engineers.
Software for performing meaningful analysis increasingly shows up on the desktops of design engineers. A finite-element model can be thought of as a system of solid springs. When a load is applied to the structure, all elements deform until all forces balance. For each element in the model, equations can be written relating displacements and forces at the nodes. The element shown here, for example, is a 2D quadrilateral having four nodes. Each node has two degrees of freedom associated with it (displacements in X and Y directions), so that the element has a total of eight degrees of freedom. There must also be a nodal force for each nodal degree, so there are also eight nodal forces for the element. These displacements and forces are identified by a coordinate numbering system for entry the computer program. For example, dxi1 is the deflection in the X direction for element i at node 1, while dyi1 is the deflection in the Y direction for the same node in the same element. Forces are identified in a similar manner, so that Fxi1 is the force in the X direction for element i at node 1. An equation relating displacements and forces for the element takes the form of basic spring equation, F = kd. For four nodes:
• k11dxi1 + k12dyi1 + k13dxi2 + k14dyi2 + . . . + k18dyi4 = Fxi1

• k21dxi1 + k22dyi1 + k23dxi2 + k24dyi2 + . . . + k28dyi4 = Fyi1

• k31dxi1 + k32dyi1 + k33dxi2 + k34dyi2 + . . . + k38dyi4 = Fxi2

• k81dxi1 + k82dyi1 + k83dxi2 + k84dyi2 + . . . + k88dyi4 = Fyi4

The k factors are stiffness coefficients relating the nodal deflections and forces, and are calculated by the finite-element program from material properties such as Young's modulus and Poisson's ratio, and from the element geometry. Thus, in the example, coefficient k13 relates deflection 3 and force 1. If degrees of freedom and nodal forces are consecutively numbered (dxi1 = d1, dyi1 = d2, Fxi1 = F1, Fyi1 = F2, and so forth), the matrix can be renumbered to show how stiffness coefficients relate nodal forces and deflections. When a structure is modeled, individual sets of matrix equations are automatically generated for each element. The elements in the model share common nodes so individual sets of matrix equations can be combined into a global set of matrix equations. This global set relates all the nodal degrees of freedom to the nodal forces, and the nodal degrees of freedom are solved simultaneously from the global matrix. When displacements for all nodes are known, the state of deformation of each element is known. And, when deformation of each element is know, the stress and strain within the element are also known. For simple static analysis, the finite-element method is a two-step process. Nodal displacements are first simultaneously calculated from the element stiffness and the nodal forces, both internal and external. Next, stresses are calculated, generally at the each element's centroid. Because displacements are calculated for only a finite number of points in the structure, the finite-element method is a numerical approximation rather than an exact solution.

Flexible Power and Energy Systems for the Evolving Factory

Aug. 29, 2024
Exploring industrial drives, power supplies, and energy solutions to reduce peak power usage and installation costs, & to promote overall system efficiency

Advancing Automation with Linear Motors and Electric Cylinders

Aug. 28, 2024
With SEW‑EURODRIVE, you get first-class linear motors for applications that require direct translational movement.

Gear Up for the Toughest Jobs!

Aug. 28, 2024
Check out SEW-EURODRIVEs heavy-duty gear units, built to power through mining, cement, and steel challenges with ease!

Flexible Gear Unit Solutions for Tough Requirements

Aug. 28, 2024
Special gear units to customer-specific requirements – thanks to its international production facilities, SEW-EURODRIVE can also build special gear units to meet customer needs...